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Eighty years ago, Ramanujan conjectured and proved some striking congruences for the partition function modulo powers of 5, 7, and 11. Until recently, only a handful of further such congruences were known. Here we report that such congruences are much more widespread than was previously known, and we describe the theoretical framework that appears to… (More)

- Scott Ahlgren, Ken Ono, KEN ONO
- 2004

If p is prime, then let φ p denote the Legendre symbol modulo p and let p be the trivial character modulo p. « p be the Gaussian hypergeometric series over F p. For n > 2 the non-trivial values of n+1 F n (x) p have been difficult to obtain. Here we take the first step by obtaining a simple formula for 4 F 3 (1) p. As a corollary we obtain a result… (More)

- MODULO, SCOTT AHLGREN, MATTHEW BOYLAN

If F (z) is a newform of weight 2λ and D is a fundamental discriminant, then let L(F ⊗ χ D , s) be the usual twisted L-series. We study the algebraic parts of the central critical values of these twisted L-series modulo primes. We show that if there are two D (subject to some local conditions) for which the algebraic part of L(F ⊗ χ D , λ) is not 0 (mod),… (More)

- Scott Ahlgren, Ken Ono
- 2005

We investigate divisibility properties of the traces and Hecke traces of singular moduli. In particular we prove that, if p is prime, these traces satisfy many congruences modulo powers of p which are described in terms of the factorization of p in imaginary quadratic fields. We also study generalizations of Lehner's classical congruences j(z)|U p ≡ 744… (More)

- Scott Ahlgren, Ken Ono, KEN ONO
- 2004

The Langlands program predicts that certain Calabi-Yau threefolds are modular in the sense that their L-series correspond to the Mellin transforms of weight 4 newforms. Here we prove that the L-function of the threefold given by P 4 i=1 (x i + x −1 i) = 0 is η 4 (2z)η 4 (4z), the unique normalized eigenform in S 4 (Γ 0 (8)).

A partition of the positive integer n into distinct parts is a decreasing sequence of positive integers whose sum is n, and the number of such partitions is denoted by Q(n). If we adopt the convention that Q(0) = 1, then we have the generating function ∞ n=0 Q(n)q n = ∞ n=1 (1 + q n) = 1 + q + q 2 + 2q 3 + 2q 4 + 3q 5 + ... From Euler's Pentagonal Number… (More)

- SCOTT AHLGREN, STEPHANIE TRENEER
- 2007

We study infinite families of generating functions involving the rank of the ordinary partition function, which include as special cases many of the generating functions introduced by Atkin and Swinnerton-Dyer in the 1950s. We prove that each of these generating functions is a weakly holomorphic modular form of weight 1/2 on some congruence subgroup Γ 1… (More)

- Scott Ahlgren, Ken Ono
- 2001

At first glance the stuff of partitions seems like child's play: Therefore, there are 5 partitions of the number 4. But (as happens in number theory) the seemingly simple business of counting the ways to break a number into parts leads quickly to some difficult and beautiful problems. Partitions play important roles in such diverse areas of mathematics as… (More)